The Killing: in cold blood

Just before our plane lands at Copenhagen airport, we fly over a wood. Looking down, I realise that's where the opening of The Killing was filmed. Admittedly, it was the most hackneyed scene in the whole 20-hour series, but let's not spoil the story. Bloody and terrified, teenage student Nanna Birk Larsen running from her attacker as night-lights from howling incoming planes strafe the birch trees. She never makes it. Her tortured, sexually abused body is later found in the boot of a car pulled from a canal.

Welcome to wonderful, wonderful Copenhagen. One minor thing that the hit thriller does is to shatter any twee notions of the Danish capital. Copenhagen's wonderfulness has been airbrushed from Forbrydelsen (The Killing's Danish title). The directors ignored its royal palaces, lovely parks and elegant 17th-century terraces, instead favouring damp motorways glistening under street lights, a 21st-century cityscape brooding beneath leaden skies. Maybe that's one reason The Killing has been a sensation in Britain – we feel at home in its rain-soaked north European heart. This Copenhagen looks like Birmingham, Manchester or Newcastle at their bleakest. And spoken Danish sounds sometimes like a Scouse-Glaswegian mashup.

Of course, these aren't the main reasons the show attracted such a hard-core cult following on BBC4. Mostly, viewers were captivated by the compelling characters. We couldn't tear ourselves away from Sarah Lund, the appealingly dysfunctional detective who in episode one is preparing to move to Sweden with her boyfriend. We were blind-sided by the shattering grief of Nanna's parents Pernille and Theis (I've never seen a TV drama that has dealt so unsparingly with family bereavement since the first episode of Twin Peaks, when Laura Palmer's body was found bagged and blue by the lake, and even that now seems stylised by comparison). And, most unexpectedly, The Killing's British faithful became engaged by Danish municipal politics, following idealistic mayoral candidate Troels Hartmann as the murder case threatens to derail his campaign and love life, and to bring down seemingly everyone in the town hall.

Never, quite possibly, have so many Britons gathered on a Saturday night to watch a subtitled TV drama – or timeshifted it to an otherwise boring Wednesday night. Now those of us who have been boring everybody silly about how great The Killing is will have to go cold turkey, while the rest of you really ought to catch up with it on DVD. "I think it's good for European culture if there are more and more of these kinds of exchanges," says the show's creator and head writer Søren Sveistrup. "Otherwise, especially for the UK, everything becomes Americanised."

The Killing averaged 500,000 viewers per episode on BBC4 – more than Mad Men or the Swedish-language version of Wallander managed on the same channel. The Wire averaged 600,000 viewers when shown on BBC2 two years ago, and last summer's finale of 24 on Sky 1 got the same number of viewers. So the viewing figures for The Killing are exceptional, not least as this is a four-year-old series that the BBC only brought to our screens – after sitting on its (allegedly very cheap) acquisition for several years – because (or so it seemed) they'd run out of Wallanders and needed something similarly bleak and Nordic to fill the gap.

"When The Killing was shown in Germany and the Netherlands, it did really well, but there certainly wasn't the excitement about it that there's been in Britain," says Lars Mikkelsen, who plays Troels Hartmann in the series. "In Germany they liked it as a whodunnit, but you guys seem to be more into the characters." That's true: Killing-besotted message boards are more likely to be concerned with whether Lund is a feminist role model or screw-up, rather than working out who killed Nanna Birk Larsen.

The Sunday Times described The Killing as "the new The Wire" (oh how much the PR people must have yearned for that hackneyed encomium to blurb on the DVD box set), but one of its delights is that it isn't American in mood or sensibility. True, its structure – each hour-long episode is a day in the investigation – borrows from Joel Surnow's "real-time" 24, but otherwise it's satisfyingly downbeat. There are no heroes, no sex and – in the misplaced analysis of the New York Times – no action. There is plenty of action, but little of it as frenetic as in 24, The Wire or The Shield.

"The creation of the series was in part an aggressiveness to US thrillers where everything's solved within one episode," says Sveistrup. "Also I never saw grief really in US thrillers. I never saw anything remotely touching. I wanted to change all that."

In a back-handed tribute to Sveistrup's conception of the series, US network AMC is to premiere an English-language remake of The Killing next week. Nanna Birk Larsen has become Rosie Larsen and Copenhagen has become Seattle. "We tried to embrace a lot of what we thought made it incredible, including the Nordic sensibility, the stoicism of Sarah Lund and the lack of that overtly frenetic behaviour that you're constantly seeing on American crime and police shows," Joel Stillerman, AMC's surely ironically titled senior vice president for original programming, production and digital content told the New York Times recently.

It makes you wonder why they didn't just screen the Danish original. "I don't think the original will ever screen there," says Sveistrup, who was consulted by the American retoolers. "It's unthinkable that Americans would ever screen it with subtitles."

But back to Denmark. Outside Copenhagen airport, the railway station offers me a choice. In one direction, trains go towards Ystad in nearby Sweden, where Henning Mankell's Detective Kurt Wallander oversees the sleepy town's counterintuitively high body count. The other way, they go to central Copenhagen, beat of small, Nicorette-chewing, Christmas-jumper-wearing, sleep deprived, borderline sociopathic, disturbingly affectless and scarily stary police commissioner Sarah Lund. Who wouldn't choose the latter?

Minutes later, I'm standing outside the Kongelige Teater (Royal Theatre) on Kongens Nytorv, where I'm due to meet Sofie Gråbøl, who plays Lund. She's just finishing rehearsals, but something's wrong. The posters are for My Fair Lady. Surely Lund is the last woman in the world who would succumb to a Henry Higgins makeover?

My phone rings. It's Gråbøl, who for 25 years has been one of Denmark's leading actors. She laughs at the idea that she could sing Eliza Dolittle. Sarah Lund laughs – the very idea! Turns out I'm at the wrong theatre – I should have gone to the Kongelige Skuespilhus, not the Kongelige Teater. She's at the latter, where she's rehearsing for an adaptation of Ingmar Bergman's Fanny and Alexander, which makes much more sense. As I walk along Nyhavn, Copenhagen's picturesque waterfront, tourists are lounging at pavement cafes overlooking boats on the canal, some tucking into suppers involving three types of herring.

Suddenly I see a familiar pair of eyes in the crowd. But those huge, beautiful eyes don't belong with that soft, wavy hairdo, nor with that smile, nor with the scarf that matches that jacket so well. Sarah Lund never did accessorise (unless wellies with jeans is accessorising). But this is the woman who plays Lund. We walk along the canal to the theatre. I catch a sidelong glance – she's wearing heels! You don't walk like Lund did. "No, when we were creating her I made her walk like a man. I copied the director of the first three episodes, who was a little macho in his walk." She walks ahead to show me, arms swinging, legs splayed – that recognisable gait of a man who wants the world to believe his balls are too big for his trousers. "She is very far from me," says Gråbøl, 42, with a laugh.

By the way, how do you pronounce your surname, I ask, mystified by those diverting Danish diacritics. "Gråbøl rhymes with trouble," she replies. Of course it does – just as Susan Sarandon rhymes with abandon.

We settle in the cafe of the thrilling new Kongelige Skuespilhus, with its stupendous view of Copenhagen's canals. Across the water is the new opera house, while towards the setting sun is Gråbøl's apartment building.

What drew you to playing Lund – was it the idea of playing a tough woman cop in an overwhelmingly male milieu? "Søren Sveistrup had this very loose idea. He only knew that it was going to be a female detective, and I only knew I wanted to work with him again." Was the idea to address institutional sexism in the workplace to make an overtly feminist drama? "No, we agreed that we didn't want to go there. It's not to diss the issue of a woman in a man's world but we felt that that's been done many times before."

Far from being then a Danish version of Helen Mirren's Jane Tennison in Prime Suspect, or a glam-but-centred-and-tough simulacrum of Above Suspicion's Anna Travis or CSI's Catherine Willows or The Wire's Kima Greggs, Lund is something new. "The classical story is of the working mother with a love life, a woman who wants to be something to everyone," says Gråbøl. "There would be scenes of her having sex with her boyfriend, cooking and all those usual cliches. We didn't want to do any of those things. Lund's not like that. She can't separate herself from the case.

"Some people see her as a superwoman, but she's not. She's brave and strong, but she has so many weak points she's almost crippled." Despite (maybe even because) of all that, Lund has become a role model. "I love Lund," wrote Grace Dent in the Guardian. "She's the sort of better woman I dream to be, shunting her love life, parenting-guilt and planned move to Sweden as she bids to change Denmark for the greater good."

Lund's appeal perhaps is that she's not so much a woman in a man's world as a traditionally male character in a woman's body – a maverick like Clint Eastwood's Dirty Harry and a frightening obsessive like John Wayne in The Searchers, though neither as self-righteous nor gun-happy as either. She's a loner guided by a superior intelligence who pursues her investigation ruthlessly, stepping on the toes of town hall politicos and lame-brained bosses alike, treating her male colleague Jan Meyer as part-chump and part-servant, hardly ever sparing the feelings of Nanna Birk Larsen's bereaved parents when she turns up, Columbo-like, with just one more question.

"The strength of the character is that you can't put her in a box," says Gråbøl. "I like it when there's something that's held back so a character isn't defined for you. I like that as an actor and I like it as an audience member. Lund isn't a stereotype – she's a character who came to life as we made the series. If a character is too obviously defined, you feel unemployed in the audience."

Good point: Lund is a satisfying enigma to whom we are tempted to impute all kinds of motivations. Unlike Stieg Larsson's Lisbeth Salander, Lund is given no history to explain how she got to be bracingly focused and tough.

So where did Sarah Lund come from? "All the time I was writing it I had a quote from Nietzsche going around my head," Sveistrup later tells me, "the one about the abyss." Nietzsche wrote: "He who fights with monsters should be careful least he thereby becomes a monster. When you stare at the abyss, the abyss stares back at you."

"Lund has looked into the abyss and perhaps it makes her what she is later on in the investigation," says Sveistrup. No wonder, perhaps, that the internal affairs guys think Lund has gone mad; no wonder her son, mum and boyfriend think they've lost her to her obsession with solving the case.

And then in episode 17, something genuinely upsetting happens that makes Lund seem human again.

"The actors weren't allowed to know the whole story of The Killing. We just got it in episodes. We would get scripts on Friday night still warm from the photocopier, and film on Mondays. Søren used to say he thought we gave better performances when we didn't know what was going on. It used to annoy me that he said that but maybe he's right – it kept you on your toes."

Gråbøl must have liked his methods because she signed up for another tour. The Killing's second season will be screened on BBC4 in the autumn. In it, Lund is a passport controller in a small town, but then her former boss Lennart Brix calls in her help on a murder case connected to the killing of civilians in Afghanistan.

"For me," says Gråbøl, "I felt I had unfinished business with Lund. I didn't want to let her go." She's planning to start on a third season later this year. What will it be about? "I don't know and Søren won't tell you," she says.

As Gråbøl and I talk, a tall man approaches our table. Who could this lavishly bearded stranger be? It's Lars Mikkelsen, the actor who played Troels Hartmann. I scarely recognised him in jeans and sweater – Hartmann was always suited and booted in the series.

"Remember," Mikkelsen says to Gråbøl as he sits down, "how we were going to have a thing in the series? The writers thought it would be a good idea if Troels and Sarah became romantically involved". "Yes," says Gråbøl, "but we had to fight against it because that would have been such a cliche." "A nice cliche, though," says Mikkelsen with a wink.

They both recall how big a hit Forbrydelsen was in Denmark in 2007. "It was watched by two million people every Sunday night, which doesn't sound like much but there are only six million people in Denmark," says Mikkelsen.

Mikkelsen heads off to prepare for his performance as Alceste in Molière's The Misanthrope, leaving Gråbøl to tell me how, shortly before she recorded the last episodes of the first season, she was approached by the relatives of a woman who was dying of cancer. The woman feared she wouldn't live long enough to find out who killed Nanna Birk Larsen. "So I wrote her a note and put it in an envelope. She read it and tore it into little pieces so the nurse couldn't find out who did it. The following day she died."